Problem 14 - University Physics with Modern Physics

Question

For a convex spherical mirror that has focal length \(f\) = -12.0 cm, what is the distance of an object from the mirror's vertex if the height of the image is half the height of the object?

Step-by-Step Solution

Answer

The object is 12 cm from the mirror.

Step 1: Understand the Mirror Equation

For a spherical mirror, we use the mirror equation: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. Since the mirror is convex, \( f = -12.0 \) cm.

Step 2: Relate Image and Object Heights

We know from the problem statement that the image height \( h_i \) is half of the object height \( h_o \), giving us the magnification equation: \( M = \frac{h_i}{h_o} = \frac{1}{2} \). Magnification is also represented as \( M = -\frac{d_i}{d_o} \).

Step 3: Substitute in the Magnification Equation

Since \( M = \frac{1}{2} \), we have \( \frac{1}{2} = -\frac{d_i}{d_o} \), leading to \( d_i = -\frac{1}{2}d_o \). This relationship will help us find the object distance \( d_o \).

Step 4: Solve the Mirror Equation with Substitution

Substitute \( d_i = -\frac{1}{2}d_o \) into the mirror equation: \( \frac{1}{-12} = \frac{1}{d_o} + \frac{1}{-\frac{1}{2}d_o} \). Simplify it to: \( \frac{1}{-12} = \frac{1}{d_o} - \frac{2}{d_o} = -\frac{1}{d_o} \).

Step 5: Solve for the Object Distance

From \( -\frac{1}{d_o} = \frac{1}{-12} \), we rearrange to find \( d_o = 12 \) cm. The object distance is 12 cm.

Key Concepts

Convex Mirror

Convex mirrors are a type of spherical mirror where the reflecting surface bulges outward, resembling the exterior of a sphere. These mirrors are known for creating virtual images that appear smaller and upright compared to the actual object. The interesting feature of a convex mirror is that it always forms images that are diminished and placed between the mirror and its focal point.
This occurs because the reflected rays diverge, and when extended backwards, they appear to converge at a point behind the mirror. Some common applications of convex mirrors include:

• Rearview mirrors in vehicles, offering a wider field of view
• Security mirrors in stores, allowing for broad surveillance
• Architectural features enhancing space perception

Understanding how a convex mirror works helps you appreciate its function in various daily life scenarios.

Mirror Equation

The mirror equation is a fundamental formula used to relate the object distance ( d_o ), image distance ( d_i ), and the focal length ( f ) of the mirror. This relationship is expressed as: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]This equation is applicable for both convex and concave mirrors, though it behaves differently based on the type of mirror. For convex mirrors, the focal length is given a negative value.
This distinction is important as it influences how you interpret the resulting image's characteristics, such as its location and size. To effectively use the mirror equation:

• Determine the type of mirror and assign the correct sign to the focal length
• Know the values of two out of three variables ( d_o , d_i , f ), and solve for the unknown
• Ensure all distances are measured from the mirror's vertex

In our original problem, we used this equation to find the object distance with known focal length and a derived relation between object and image distances.

Magnification

Magnification is an important concept when working with mirrors and lenses. It measures how much larger or smaller the image is compared to the object. The formula for magnification ( M ) is given by: \[ M = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\]where ( h_i ) is the image height and ( h_o ) is the object height. The negative sign in the magnification equation applies to mirrors, indicating inversion of the image in cases where the image is real. However, for convex mirrors, the images formed are virtual, upright, and diminished, making them oriented the same as the object.Key points about magnification include:

• In convex mirrors, magnification is always less than 1, indicating a smaller image
• A positive magnification signifies an upright image
• A negative magnification suggests an inverted image, which is typical with real images formed by concave mirrors

When solving problems, understanding magnification helps predict characteristics of the image like its orientation and relative size to the object.

Focal Length

Focal length is a crucial concept associated with spherical mirrors. It is the distance from the mirror's surface to its focal point, the point where parallel light rays either converge (concave mirror) or seem to diverge from (convex mirror). For spherical mirrors, the focal length ( f ) is half of the radius of curvature ( R ): \[ f = \frac{R}{2}\]This relationship helps calculate or verify focal lengths using geometry principles.
In convex mirrors, f is considered negative, which affects many calculations involving the mirror equation.Why is focal length important?

• It helps determine the image position and size using the mirror equation
• It defines the mirror's power and effectiveness at forming images
• It is essential for designing optical instruments like telescopes and cameras

In the exercise scenario, understanding the mirror's focal length was central to predicting image formation and solving for object distance.